Investigating Area With Paper Folding

Last Thursday was a half-day for students because we had Open House in the evening. I decided it was a good day for my geometry students to do some paper folding. I gave each student about 5 sheets of hamburger patty paper and these instructions:

Task 1: construct (by folding) a square that is 1/4 the area of the original square.

Big deal. Everyone got it, except Daniel, but he didn’t follow instructions, he constructed a rectangle with 1/4 the area. The kids jokingly gave him a hard time. I reminded the class that Daniel’s left arm was still in a cast, thus squares and rectangles were the same to him.

Task 2: construct a triangle that has 1/4 the area of the original square.

Again, no problems. Everyone folded the same way.

Task 3: construct another triangle, still 1/4 the area of the original, but this triangle may not be congruent to the one you created in Task 2.

Some time passed, but not too long before Slater got it, Zach did too, they shared with the class.

Task 4: construct a square that is 1/2 the area of the original square.

Easy-peasy-lemon-squeezy.

Task 5: construct a square that is 1/2 the area of the original square — so same instruction as Task 4 — however, it must be “oriented differently.”

And then I went to sit down, put my feet up (okay, figuratively speaking only), drank some water, and peacefully watched as my students slowly but surely mangled up their papers trying to create this square. They asked for more paper. I gave them more paper.

Austin: What do you mean “oriented differently”?

Me: Its orientation is different.

A: That didn’t help…

M: I’m not here to help you. Not today.

Zach, waving his folded paper above his head and proceeded to walk up to me: Oh, I got it!

M: No, you don’t got it. Sit down.

Josh: I got it!!

M: No you don’t.

J: But you didn’t even look at my paper.

M, pretending to look: There, I looked.

I watched with satisfaction as everyone struggled with Task 5.

The kids know I live for moments like this. They are stuck, and they have to think really hard.

Even Slater, a self-proclaimed math prodigy (who since then had elevated himself to just “academic prodigy”) was quiet and using his pencil to do some calculations on paper. Even Bobby — who once claimed that he and Slater should get the same IQ score if put to the test — sat quietly, arms to his sides staring at the patty paper.

The bell rang. I told them to work on it for homework, and the hint was “right triangle.”

Alex: Someone will look it up online, Mrs. Nguyen.

Me: That’s their loss.

We did Task 5 together the next day. The smiles returned to their faces. I love teaching.

I am thoroughly convinced by looking at your picture that you and your class have not solved #5, as your squares are not 50% of the original square’s area. You have folded in 7/16ths, leaving a square which is 9/16th the original size.